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x---->

<---y

www.PROJEKTOMANIA.com Temat = 13

α = 0.32

β = 0.6

γ = 1.28

2

L = 5.22

m

EJ = 1128 kNm

x = α

x = 0.32

y = 1 − α = .68

1

2

M

= M − ⋅ ⋅

A

MB

q L

B

2

1. Postac wyboczenia 1.1 Równanie rózniczkowe osi odksztalconej Przedzial II

Przedzial I

0 < x < α ⋅L

0 < y < L − α ⋅L

1

2

M

=

+

⋅ − ⋅ ⋅

1

y(x)

MA VA x

q x

2

2

M

=

− ⋅ ⋅

y(y)

MB

q y

2

2

EJ⋅

1

w''

=

⋅ ⋅

−

⋅ −

I(x)

q x

V

M

2

2

A x

A

β⋅EJ⋅

1

w''

=

⋅ ⋅

−

II(y)

q y

M

2

B

3

2

EJ⋅

1

w'

=

⋅ ⋅

1

−

⋅

−

⋅ +

I(x)

q x

V

M

C

3

6

2 A x

A x

1

β⋅EJ⋅

1

w'

=

⋅ ⋅

−

⋅ +

II(y)

q y

M

C

6

B y

2

4

3

1

2

EJ⋅

1

w

=

⋅ ⋅

1

− ⋅

⋅

− ⋅

⋅

+

⋅ +

I(x)

q x

V

x

M

x

C

D

4

2

24

6

A

2

A

1 x

1

β⋅EJ⋅

1

w

=

⋅ ⋅

1

− ⋅

⋅

+

⋅ +

II(y)

q y

M

y

C

D

24

2

B

2 y

2

1.2 Wyznaczenie stalych z warunków brzegowych 1) w (0) = 0

I

2) w' (0) = 0

I

3) w` (0) = 0

II

4) w' (αL) = -w' (L-αL) I

II

5) w (αL) =w (L-αL) I

II

2

3

3

M

= −

⋅

=

⋅

=

⋅

A

0.3613

q L

C1 0

q L

C2 0

q L

4

V

=

⋅

=

⋅

A

1

q L

D1 0

q L

4

D =

⋅

2

0.03125

q L

2

M

=

⋅

B

0.1387

q L

Wynik kontrolny w'(αL): 2

1

⋅

⋅

3

2

3

2

q⋅

1

x −

V ⋅

− M ⋅ +

1

C

=

⋅q⋅

1

x −

⋅1⋅x − (−0.3613)⋅x + 0 =

q L

0.06988

Kontrola = 0.06987

6

2 A x

A x

1

6

2

EJ

2. Obciazenie krytyczne 2.1 Kryterium energetyczne Timoshenki

⌠

⌠

⎮

2

⎮

2

EJ⋅w'' dx

⎮

+

N⋅w'

x

⎮

d = 0

⌡

⌡

I)

1 ⎛ 1

3

2

2

⎞

1 ⎡1

3

2

2

⎤

w'

=

⋅⎜ ⋅ ⋅

1

− ⋅

⋅ ⋅ ⋅ −

⋅ ⋅ ⋅

+

⎟ =

⋅⎢ ⋅ ⋅

1

− ⋅ ⋅ ⋅ ⋅ − (−

⋅ ⋅ ⋅

+ ⎥

I(x)

q x

VA x q L MA x q L

C1

q x

1 x q L

0.3613) x q L

0

EJ ⎝ 6

2

⎠

EJ ⎣6

2

⎦

1 ⎛ 1

2

2⎞

1 ⎡1

2

2⎤

w''

=

⋅⎜ ⋅ ⋅ −

⋅ ⋅ ⋅ −

⋅ ⋅ ⎟ =

⋅⎢ ⋅ ⋅ − ⋅ ⋅ ⋅ − (−

⋅ ⋅ ⎥

I(x)

q x

VA x q L MA q L

q x

1 x q L

0.3613) q L

EJ ⎝ 2

⎠

EJ ⎣2

⎦

N x = (− − γ ⋅

= [(−

−

⋅

I(x)

1

) P

1)

1.28] P

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www.PROJEKTOMANIA.com II)

1

⎛ 1

3

⎞

1

⎛ 1

3

⎞

w'

=

⋅⎜ ⋅ ⋅ −

⋅ +

⎟ =

⋅⎜ ⋅ ⋅ −

⋅ + ⎟

II(x)

q y

M

C

q y

0.1387 y

0

β⋅

B y

2

EJ ⎝ 6

⎠

0.6⋅EJ ⎝ 6

⎠

1

⎛ 1

2

⎞

1

⎛ 1

2

⎞

w''

=

⋅⎜ ⋅ ⋅ −

⎟ =

⋅⎜ ⋅ ⋅ −

⎟

II(x)

q y

M

q y

0.1387

β⋅

B

EJ ⎝ 2

⎠

0.6⋅EJ ⎝ 2

⎠

N

= −

II(x)

P

2.2 Wartosc krytyczna sily P

α⋅L

⌠

L−α⋅L

α⋅L

L−α⋅L

⎮

⌠

⌠

⌠

2

⎮

2

⎮

2

⎮

2

EJ⋅w''

dx

⎮

+

β⋅ ⋅

− ⋅

(

+ ⋅

− ⋅

=

I

EJ w''

y

d

P

1

γ) w'

x

d

P

w'

y

d

0

⌡

⎮

II

⌡

⎮

I

⌡

⎮

II

⌡

0

0

0

0

0.32⋅L

⌠

L−0.32⋅L

0.32⋅L

L−0.32⋅L

⎮

⌠

⌠

⌠

2

⎮

2

⎮

2

⎮

2

=

EJ⋅w''

x

⎮

+

⋅ ⋅

− ⋅

⋅

− ⋅

I d

.6 EJ w''

y

d

P

2.28 w'

dx

P

w'

y

d

⌡

⎮

II

⌡

⎮

I

⌡

⎮

II

⌡

0

0

0

0

α⋅L

⌠

0.32⋅L

⌠

⎮

⎡

2

⎮

2

1 ⎛ 1

2

⎞⎤

1 ⎛ 1

2

⎞

I

= ⎮

⋅⎢ ⎜ ⋅ ⋅ −

⋅ ⋅ ⋅ −

⋅ ⋅ ⎟⎥

= ⎮

⋅⎜ ⋅ ⋅

− ⋅ ⋅ +

⋅

⎟

1

EJ

q x

V

q L

M

L

x

A x

A x

d

q x

x q L

.3613 x L

⋅

x

d

⎮

⎣EJ ⎝ 2

⎠⎦

⎮

⌡

EJ ⎝ 2

⎠

⌡

0

0

2

5

⋅L

J =

q

1

0.01719

EJ

L−α⋅L

⌠

L−0.32⋅L

⌠

⎮

⎡

2

⎮

2

1

⎛ 1

2

⎞⎤

1.6666666666666666667 ⎛ 1

2

⎞

I

= ⎮

β⋅ ⋅⎢

⋅⎜ ⋅ ⋅ −

⎟⎥

= ⎮

⋅⎜ ⋅ ⋅

−

⎟

2

EJ

q y

M

y

d

q y

.1387

dy

⎮

⎣β⋅

B

EJ ⎝ 2

⎠⎦

⎮

⌡

EJ

⎝ 2

⎠

⌡

0

0

2

5

− 3

q ⋅L

J =

×

2

9.6901

10

EJ

α⋅L

⌠

0.32⋅L

⌠

⎮

⎡

2

2

1 ⎛ 1

3

1

2

⎞⎤

⎮

2.28 ⎛ 1

3

1

2

⎞

I

= ⎮

(

+ ⋅⎢ ⋅⎜ ⋅ ⋅ − ⋅

⋅ ⋅ ⋅ −

⋅ ⋅ +

⎟⎥

= ⎮

⋅⎜ ⋅ ⋅

− ⋅ ⋅

⋅ +

⋅

⎟

3

1

γ)

q x

V

x q L

M

L

C

x

A

A x

1

d

q x

q x L

.3613 x L

⋅

x

d

⎮

⎣EJ ⎝ 6

2

⎠⎦

⌡

2

⎮

⎝ 6

2

⎠

EJ

0

⌡0

2

7

− 3

q ⋅L

J =

×

3

1.59332

10

2

EJ

L−α⋅L

⌠

L−0.32⋅L

⌠

⎮

⎡

2

2

1

⎛ 1

3

⎞⎤

⎮

2.7777777777777777779 ⎛ 1

3

⎞

I

= ⎮

⎢

⋅⎜ ⋅ ⋅ −

⋅ +

⎟⎥

= ⎮

⋅⎜ ⋅ ⋅

−

⋅ ⎟

4

q y

M

C

dy

q y

.1387 y

y

d

⎮

⎣β⋅

B y

2

EJ ⎝ 6

⎠⎦

⌡

2

⎮

⎝ 6

⎠

EJ

0

⌡0

2

7

− 3

q ⋅L

J =

×

4

2.60749

10

2

EJ

J +

1

J2

EJ

P

=

=

k

6.3989

J +

3

J4

2

L

EJ

P

=

⋅

=

kr

Pk

264.895

kN

2

L

2.3 Wspólczynnik wyboczeniowy π2⋅

μ

EJ

=

μ = 1.24193

2

P ⋅

kr L

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